Simplicial Sets

So I’ve been reading this introduction to simplicial sets, because I’m at a point of needing to understand more of the model theory of homotopy type theory and I haven’t touched algebraic topology in any form for over a decade.

An obvious question one might ask is “why learn this?” and, honestly, I don’t entirely understand the reason yet other than that I want to actually read the papers on the model theory of HoTT and have them make any sense whatsoever. Of course, there’s a larger question of “why care about HoTT/higher dimensional type theories” and I’ll hopefully soon have a cogent explanation of why I care if not a moving plea for others to do the same, one that doesn’t simply rely on “math is purdy”.

The paper starts off smoothly with the idea of abstract simplicial complexes which are just a nat-indexed family of sets. We have that the family X^n is a simplicial complex when, for each j, X^{j} consists of size j+1 subsets of X^0, which is the set of vertices, such that all the subsets of an element of X^j are elements of one of the prior sets in the sequence.

For example, if we have X^0 = [0,1,2,3], X^1 = [[0,1],[0,2],[1,2],[0,3]] and X^2 = [ [0,1,2] ] then this is a simplicial complex describing a triangle with an extra leg sticking out not connected to a surface. We can see that this example satisfies the subset property.

The basic picture for these are that each level describes the points, the lines, the surfaces, the volumes, etc. and the subset condition means that if you have a surface then the lines must be in the complex, and if you have a line then the vertices must be in the complex, etc. It seems straight forward to me now, but for some reason when I was first reading this I kept making it orders of magnitude more complicated than it needed to be. He doesn’t describe it in terms of all subsets in his definition, but I find that much easier to picture in my head.

Simplicial maps are really then just maps from one set of vertices to another, and then the action on the entire complex is generated by the action on vertices.

As another example, let’s take our X above and the complex Y^0 = [0,1,2], Y^1 = [[0,1],[1,2],[2,3]], Y^2 = [ [0,1,2] ]. Now we can let f^0 be the identity on 0,1, and 2 and map 3 to 0. Then we actually will be collapsing the line [0,3] down to the point [0] and “shrinking” that dangling bit off the triangle to just a point.

The author goes on to explain that we can tighten up the definition of a simplicial complex by insisting that vertices always be (strictly) ordered. He calls these, unsurprisingly, ordered simplices. We don’t lose any power with this restriction, but we gain the ability to describe complexes in terms of maps over a canonical ordered complex.

Now we get to something cute: face maps. Since every set in X^j has j+1 elements, this means that we can have a family of maps d_i : X^j \to X^{j-1} for i \in [0..n], what these maps do is “delete” the ith element of the subset. This family of maps essentially generates all the possible subsets of elements in X^j, so we really just need to consider these maps and their compositions in order to easily categorize the content of a complex. We also have the algebraic identity that d_i \circ d_j = d_{j-1} d_i when i < j. Of course, whenever we have algebraic identities that characterize something that’s a good clue that there’s a generalization, right?

That’s how we come to the topic of Delta sets and Delta maps. A Delta set is like a more abstracted abstract simplicial complex, where the only conditions we have are that X is a family of sets equipped with a family of maps that follow the same identity as the face maps. The author goes on to give examples of how Delta sets are more general than simplicial complexes because you can, in principle, describe things a cone or a pair of simplices that have two distinct edges between them.

Now gets to the beginning of the exciting stuff for me: we can think of every Delta set as coming from a functor from the simplicial-complex \Delta to the category of sets, i.e. \Delta^{op} \to Set, where \Delta is the complex where \Delta^n = [n] so that \Delta^1 = [0,1], \Delta^2 = [0,1,2] etc. Now the maps of this category are going to be the strictly order preserving functions \Delta^n \to \Delta^m (which of course means that n<m), but these can be generated by the inclusions D_i : \Delta^n \to \Delta^{n+1} where these inclusions are the ones that map onto every point, in order, skipping i. Now when we take the op of the category, utterly shockingly these maps become the d_i face maps because they actually obey the identities in the following way: d_i \circ d_j = d_{j-1} \circ d_i when i < j becomes D_j \circ D_i = D_i \circ D_{j-1} which should work because for, say j=1 and i=0 then we have that since D^1_0 sends [0,1] to [1,2] and D^2_1 sends [0,1,2] to [0,2,3] and so the composition will be [0,1] \to [2,3] and the other direction will first take [0,1] to [1,2] and then [0,1,2] to [1,2,3] which means that the composition will be, again, [2,3]. I know that’s hardly a proof but it does show that at least in one concrete example the inclusions work correctly. Since the laws we want are forced by saying that the map from \Delta to Set must be a functor, this gives us a new way to describe Delta sets.

Next up, we hit the section actually describing simplicial sets and degeneracy maps.

This is really all I feel like writing today of my notes on simplicial sets, but this is only about a quarter of the introduction. So we’ll cover simplicial sets, realization, and simplicial objects next time.


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